Geometric bijections of graphs and regular matroids
Yuen, Chi Ho
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The Jacobian of a graph, also known as the sandpile group or the critical group, is a finite group abelian group associated to the graph; it has been independently discovered and studied by researchers from various areas. By the Matrix-Tree Theorem, the cardinality of the Jacobian is equal to the number of spanning trees of a graph. In this dissertation, we study several topics centered on a new family of bijections, named the geometric bijections, between the Jacobian and the set of spanning trees. An important feature of geometric bijections is that they are closely related to polyhedral geometry and the theory of oriented matroids despite their combinatorial description; in particular, they can be generalized to Jacobians of regular matroids, in which many previous works on Jacobians failed to generalize due to the lack of the notion of vertices. Topics discussed in the dissertation include: (i) the combinatorics of break divisors and the ABKS decomposition, (ii) group actions of the Jacobian on the circuit-cocircuit reversal system, (iii) geometric bijections and their connections with Ehrhart theory and algorithms, (iv) Bernardi processes and their relations with geometric bijections. We also provide converses and extensions of several main results.